Math Core

You already know that the area of a rectangle is length times width. But what if a garden bed is $\dfrac{3}{4}$ of a meter long and $\dfrac{2}{3}$ of a meter wide? You can't count whole square meters. In this lesson you will see that the same rule still works: multiply the side lengths, even when they are fractions.

Tiling with small squares

Start with a square that is 1 unit long on each side. Its area is 1 square unit. Cut it into 4 equal columns and 3 equal rows. That makes $4 \times 3 = 12$ equal tiles, so each tile has an area of $\dfrac{1}{12}$ square unit. Each tile is $\dfrac{1}{4}$ unit wide and $\dfrac{1}{3}$ unit tall.

Now shade a rectangle that is $\dfrac{3}{4}$ unit wide and $\dfrac{2}{3}$ unit tall.

A 1-by-1 square cut into 12 tiles. The shaded rectangle is 3 tiles wide and 2 tiles tall.

The shaded rectangle is 3 tiles wide and 2 tiles tall, so it covers $3 \times 2 = 6$ tiles. Its area is $\dfrac{6}{12} = \dfrac{1}{2}$ square unit. Now multiply the side lengths:

\frac{3}{4} \times \frac{2}{3} = \frac{3 \times 2}{4 \times 3} = \frac{6}{12}

Same answer. The numerators count the tiles in the rectangle, and the denominators count the tiles in the whole square.

Area of a rectangle

For any rectangle, even one with fraction or mixed number sides:

\text{Area} = \text{length} \times \text{width}

The answer is in square units.

Worked example: Fraction sides

A tile is $\dfrac{3}{5}$ of a meter long and $\dfrac{1}{2}$ of a meter wide. What is its area?

\frac{3}{5} \times \frac{1}{2} = \frac{3}{10}

The area is $\dfrac{3}{10}$ square meter.

Mixed number sides

When the sides are mixed numbers, you can change them to fractions and multiply. You can also split the rectangle into parts, just like the area model for whole numbers.

Worked example: Split into four parts

Find the area of a rectangle that is $2\dfrac{1}{2}$ feet long and $1\dfrac{1}{3}$ feet wide.

Split the length into $2 + \dfrac{1}{2}$ and the width into $1 + \dfrac{1}{3}$ :

The four parts have areas 2, 1/2, 2/3 and 1/6 square feet.

Find each part:

$2 \times 1 = 2$
$\dfrac{1}{2} \times 1 = \dfrac{1}{2}$
$2 \times \dfrac{1}{3} = \dfrac{2}{3}$
$\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}$

Add them, using sixths: $2 + \dfrac{3}{6} + \dfrac{4}{6} + \dfrac{1}{6} = 2 + \dfrac{8}{6} = 3\dfrac{2}{6} = 3\dfrac{1}{3}$ square feet.

Check by multiplying fractions: $\dfrac{5}{2} \times \dfrac{4}{3} = \dfrac{20}{6} = 3\dfrac{1}{3}$ . It matches.

Worked example: A garden

A garden is 4 yards long and $2\dfrac{3}{4}$ yards wide. What is its area?

4 \times 2\frac{3}{4} = \frac{4}{1} \times \frac{11}{4} = \frac{44}{4} = 11

The area is 11 square yards. Check with parts: $4 \times 2 = 8$ and $4 \times \dfrac{3}{4} = 3$ , and $8 + 3 = 11$ .

Common mistake

Don't multiply only the whole numbers and only the fractions. $2\dfrac{1}{2} \times 1\dfrac{1}{3}$ is not $2 + \dfrac{1}{6}$ . The picture shows why: that misses the two middle parts, $\dfrac{1}{2}$ and $\dfrac{2}{3}$ .

Practice

Practice 1

A rectangle is $\dfrac{1}{2}$ inch long and $\dfrac{1}{4}$ inch wide. What is its area in square inches?

Type a number.

Practice 2

A photo is $\dfrac{3}{4}$ foot long and $\dfrac{2}{3}$ foot wide. What is its area in square feet?

Type a number.

Practice 3

A unit square is cut into tiles that are each $\dfrac{1}{5}$ unit wide and $\dfrac{1}{3}$ unit tall. How many of these tiles cover a rectangle that is $\dfrac{4}{5}$ unit wide and $\dfrac{2}{3}$ unit tall?

Type a number.

Practice 4

What is the area of a rectangle that is $\dfrac{5}{6}$ meter long and $\dfrac{2}{5}$ meter wide?

Practice 5

A shelf is 5 feet long and $\dfrac{3}{4}$ foot deep. What is the area of its top, in square feet?

Type a number.

Practice 6

A rectangle is $1\dfrac{1}{2}$ meters long and $\dfrac{2}{3}$ meter wide. What is its area in square meters?

Type a number.

Practice 7

A rug is $3\dfrac{1}{2}$ feet long and $2\dfrac{1}{4}$ feet wide. What is its area in square feet?

Type a number.

Practice 8

Garden A is $\dfrac{3}{4}$ yard by 2 yards. Garden B is $1\dfrac{1}{2}$ yards by $1\dfrac{1}{2}$ yards. How many more square yards is Garden B than Garden A?

Type a number.